Kim, Hyuk Complete left-invariant affine structures on nilpotent Lie groups. (English) Zbl 0591.53045 J. Differ. Geom. 24, 373-394 (1986). Which simply connected Lie groups admit a complete left-invariant affine structure, or equivalently which Lie groups act simply transitively on \({\mathbb{R}}^ n\) as affine transformations, is an important open question in the study of affine manifolds. We study some basic properties of such Lie groups through the left-symmetric product on its Lie algebra defined by the connection. The classification problem of such structures is known for dimensions less than 4. The case of dimension 4 when the group is nilpotent is carried out in this paper. As an application of this classification one can easily determine all the possible left-invariant complete flat pseudo-Riemannian structures on the simply connected nilpotent Lie groups of dimension \(\leq 4.\) This classification especially reveals that there are two isomorphism classes of complete left-invariant flat structures such that they do not contain any pure translations when they are viewed as simple transitive actions on \({\mathbb{R}}^ n\) by affine transformations. Cited in 2 ReviewsCited in 63 Documents MSC: 53C30 Differential geometry of homogeneous manifolds 22E25 Nilpotent and solvable Lie groups 53B05 Linear and affine connections Keywords:left-symmetric algebra; simply connected Lie groups; left-invariant affine structure; affine manifolds; pseudo-Riemannian structures; flat structures PDFBibTeX XMLCite \textit{H. Kim}, J. Differ. Geom. 24, 373--394 (1986; Zbl 0591.53045) Full Text: DOI